The Rees matrix semigroups over 0-group and generalized Rees matrix semigroups were developed by appling the concept of R－left cancellative monoid.The normal Rees matrix semigroups over R－left cancellative monoid was introduced.Its nature and structure were discussed and it is provd that the normal Rees matrix semigroup over R－left cancellative monoid is primitively decomposable wrpp semigroup,and is also completely 0－J^**－simple wrpp semigroup.These results partly extend the conclusion about the nature and structure of Rees matrix semigroups over monoids,cancellative monoids and left cancellative semigroups.